Olympiad problems

2017 ELMO Shortlist, 1

Tags: algebra

Let $0<k<\frac{1}{2}$ be a real number and let $a_0, b_0$ be arbitrary real numbers in $(0,1)$. The sequences $(a_n)_{n\ge 0}$ and $(b_n)_{n\ge 0}$ are then defined recursively by $$a_{n+1} = \dfrac{a_n+1}{2} \text{ and } b_{n+1} = b_n^k$$ for $n\ge 0$. Prove that $a_n<b_n$ for all sufficiently large $n$. [i]Proposed by Michael Ma

2018 CCA Math Bonanza, L5.2

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Two circles of equal radii are drawn to intersect at $X$ and $Y$. Suppose that the two circles bisect each other's areas. If the measure of minor arc $\widehat{XY}$ is $\theta$ degrees, estimate $\left\lfloor1000\theta\right\rfloor$. An estimate of $E$ earns $2e^{-\frac{\left|A-E\right|}{50000}}$ points, where $A$ is the actual answer. [i]2018 CCA Math Bonanza Lightning Round #5.2[/i]

2020 MBMT, 9

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Consider a regular pentagon $ABCDE$, and let the intersection of diagonals $\overline{CA}$ and $\overline{EB}$ be $F$. Find $\angle AFB$. [i]Proposed by Justin Chen[/i]

2011 Czech and Slovak Olympiad III A, 1

Tags: geometry

In a certain triangle $ABC$, there are points $K$ and $M$ on sides $AB$ and $AC$, respectively, such that if $L$ is the intersection of $MB$ and $KC$, then both $AKLM$ and $KBCM$ are cyclic quadrilaterals with the same size circumcircles. Find the measures of the interior angles of triangle $ABC$.

2019 Durer Math Competition Finals, 15

Tags: diophantine , diophantine equation , number theory , sum

The positive integer $m$ and non-negative integers $x_0, x_1,..., x_{1001}$ satisfy the following equation: $$m^{x_0} =\sum_{i=1}^{1001}m^{x_i}.$$ How many possibilities are there for the value of $m$?

2006 Thailand Mathematical Olympiad, 6

Tags: algebra , inequalities

Let $a, b, c$ be positive reals. Show that $$1 +\frac{3}{ab + bc + ca}\ge \frac{6}{a + b + c}$$

2008 Swedish Mathematical Competition, 1

Tags: convex , geometry , rhombus , inscribed

A rhombus is inscribed in a convex quadrilateral. The sides of the rhombus are parallel with the diagonals of the quadrilateral, which have the lengths $d_1$ and $d_2$. Calculate the length of side of the rhombus , expressed in terms of $d_1$ and $d_2$.

2019 AMC 10, 3

Tags: percent

In a high school with $500$ students, $40\%$ of the seniors play a musical instrument, while $30\%$ of the non-seniors do not play a musical instrument. In all, $46.8\%$ of the students do not play a musical instrument. How many non-seniors play a musical instrument? $\textbf{(A) } 66 \qquad\textbf{(B) } 154 \qquad\textbf{(C) } 186 \qquad\textbf{(D) } 220 \qquad\textbf{(E) } 266$

1999 USAMTS Problems, 2

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The Fibonacci numbers are defined by $F_1=F_2=1$ and $F_n=F_{n-1}+F_{n-2}$ for $n>2$. It is well-known that the sum of any $10$ consecutive Fibonacci numbers is divisible by $11$. Determine the smallest integer $N$ so that the sum of any $N$ consecutive Fibonacci numbers is divisible by $12$.

Ukraine Correspondence MO - geometry, 2007.11

Tags: ratio , circumcircle , geometry

Denote by $B_1$ and $C_1$, the midpoints of the sides $AB$ and $AC$ of the triangle $ABC$. Let the circles circumscribed around the triangles $ABC_1$ and $AB_1C$ intersect at points $A$ and $P$, and let the line $AP$ intersect the circle circumscribed around the triangle $ABC$ at points $A$ and $Q$. Find the ratio $\frac{AQ}{AP}$.

2016 Brazil Team Selection Test, 2

Tags: functional equation

Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.

2024 Korea National Olympiad, 1

Tags: geometry

Let there be a circle with center $O$, and three distinct points $A, B, X$ on the circle, where $A, B, O$ are not collinear. Let $\Omega$ be the circumcircle of triangle $ABO$. Segments $AX, BX$ intersect $\Omega$ at points $C(\neq A), D(\neq B)$, respectively. Prove that $O$ is the orthocenter of triangle $CXD$.

1983 Miklós Schweitzer, 10

Tags: algebra , function , domain , advanced fields , geometry

Let $ R$ be a bounded domain of area $ t$ in the plane, and let $ C$ be its center of gravity. Denoting by $ T_{AB}$ the circle drawn with the diameter $ AB$, let $ K$ be a circle that contains each of the circles $ T_{AB} \;(A,B \in R)$. Is it true in general that $ K$ contains the circle of area $ 2t$ centered at $ C$? [i]J. Szucs[/i]

2020 Romanian Master of Mathematics Shortlist, C1

Tags: combinatorics , grid

Bethan is playing a game on an $n\times n$ grid consisting of $n^2$ cells. A move consists of placing a counter in an unoccupied cell $C$ where the $2n-2$ other cells in the same row or column as $C$ contain an even number of counters. After making $M$ moves Bethan realises she cannot make any more moves. Determine the minimum value of $M$. [i]United Kingdom, Sam Bealing[/i]

2009 QEDMO 6th, 11

Tags: incircle , geometry , concurrency

The inscribed circle of a triangle $ABC$ has the center $O$ and touches the triangle sides $BC, CA$ and $AB$ at points $X, Y$ and $Z$, respectively. The parallels to the straight lines $ZX, XY$ and $YZ$ the straight lines $BC, CA$ and $AB$ (in this order!) intersect through the point $O$. Points $K, L$ and $M$. Then the parallels to the straight lines $CA, AB$ and $BC$ intersect through the points $K, L$ and $M$ in one point.

2022 Thailand Mathematical Olympiad, 5

Tags: function , functional equation , geometry , analytic geometry

Determine all functions $f:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ that satisfies the equation $$f\left(\frac{x+y+z}{3},\frac{a+b+c}{3}\right)=f(x,a)f(y,b)f(z,c)$$ for any real numbers $x,y,z,a,b,c$ such that $az+bx+cy\neq ay+bz+cx$.

2023 South East Mathematical Olympiad, 1

Tags: algebra , sequence

The positive sequence $\{a_n\}$ satisfies:$a_1=1$ and $$a_n=2+\sqrt{a_{n-1}}-2 \sqrt{1+\sqrt{a_{n-1}}}(n\geq 2)$$ Let $S_n=\sum\limits_{k=1}^{n}{2^ka_k}$. Find the value of $S_{2023}$.

1969 Canada National Olympiad, 8

Tags: function , induction , strong induction , algebra

Let $f$ be a function with the following properties: 1) $f(n)$ is defined for every positive integer $n$; 2) $f(n)$ is an integer; 3) $f(2)=2$; 4) $f(mn)=f(m)f(n)$ for all $m$ and $n$; 5) $f(m)>f(n)$ whenever $m>n$. Prove that $f(n)=n$.

KoMaL A Problems 2022/2023, A. 831

Tags: geometry

In triangle $ABC$ let $F$ denote the midpoint of side $BC$. Let the circle passing through point $A$ and tangent to side $BC$ at point $F$ intersect sides $AB$ and $AC$ at points $M$ and $N$, respectively. Let the line segments $CM$ and $BN$ intersect in point $X$. Let $P$ be the second point of intersection of the circumcircles of triangles $BMX$ and $CNX$. Prove that points $A, F$ and $P$ are collinear. Proposed by Imolay András, Budapest

2010 Romania Team Selection Test, 3

Tags: rectangle , geometry

Two rectangles of unit area overlap to form a convex octagon. Show that the area of the octagon is at least $\dfrac {1} {2}$. [i]Kvant Magazine [/i]

2007 Princeton University Math Competition, 2

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Find the largest integer $n$ which equals the product of its leading digit and the sum of its digits.

2022 Bosnia and Herzegovina Junior BMO TST, 2

Tags: number theory

Let $a,b,c$ be positive integers greater than $1$ such that $$p=ab+bc+ac$$ is prime. A) Prove that $a^2, b^2, c^2$ all have different reminder $mod\ p$. B) Prove that $a^3, b^3, c^3$ all have different reminder $mod\ p$.

2023 BAMO, 4

Tags: combinatorics

Zaineb makes a large necklace from beads labeled $290, 291, \ldots, 2023$. She uses each bead exactly once, arranging the beads in the necklace any order she likes. Prove that no matter how the beads are arranged, there must be three beads in a row whose labels are the side lengths of a triangle.

2022/2023 Tournament of Towns, P3

Tags: geometry , pentagon , angle chasing

A pentagon $ABCDE$ is circumscribed about a circle. The angles at the vertices $A{}$, $C{}$ and $E{}$ of the pentagon are equal to $100^\circ$. Find the measure of the angle $\angle ACE$.

2004 Postal Coaching, 10

Tags: geometric transformation , reflection , perpendicular bisector , angle bisector , geometry

A convex quadrilateral $ABCD$ has an incircle. In each corner a circle is inscribed that also externally touches the two circles inscribed in the adjacent corners. Show that at least two circles have the same size.